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19
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4answers
957 views

When can a global symmetry be gauged?

Take a classical field theory described by a local Lagrangian depending on a set of fields and their derivatives. Suppose that the action possesses some global symmetry. What conditions have to be ...
19
votes
1answer
787 views

Why does charge conservation due to gauge symmetry only hold on-shell?

While deriving Noether's theorem or the generator(and hence conserved current) for a continuous symmetry, we work modulo the assumption that the field equations hold. Considering the case of gauge ...
16
votes
2answers
433 views

Coulomb gauge fixing and “normalizability”

The Setup Let Greek indices be summed over $0,1,\dots, d$ and Latin indices over $1,2,\dots, d$. Consider a vector potential $A_\mu$ on $\mathbb R^{d,1}$ defined to gauge transform as $$ A_\mu\to ...
14
votes
4answers
1k views

To which extent is general relativity a gauge theory?

In quantum mechanics, we know that a change of frame -- a gauge transform -- leaves the probability of an outcome measurement invariant (well, the square modulus of the wave-function, i.e. the ...
13
votes
2answers
647 views

Noether's theorem and gauge symmetry

I'm confused about Noether's theorem applied to gauge symmetry. Say we have $$\mathcal L=-\frac14F_{ab}F^{ab}.$$ Then it's invariant under $A_a\rightarrow A_a+\partial_a\Lambda.$ But can I say that ...
13
votes
2answers
575 views

If gauge symmetries are fake, then why do we care if they are anomalous?

My understanding is that gauge symmetries are fake in that they are only redundancies of our description of the system that we put in (either knowingly or unknowingly) see Gauge symmetry is not a ...
12
votes
1answer
503 views

What is a “free” non-Abelian Yang-Mill's theory?

I hope this question will not be closed down as something completely trivial! I did not think about this question till in recent past I came across papers which seemed to write down pretty much ...
11
votes
1answer
328 views

**Group structure** in Chern-Simons theory?

A non-Abelian Chern-Simons(C-S) has the action $$ S=\int L dt=\int \frac{k}{4\pi}Tr[\big( A \wedge d A + (2/3) A \wedge A \wedge A \big)] $$ We know that the common cases, $A=A^a T^a$ is the ...
10
votes
2answers
705 views

Are there massless bosons at scales above electroweak scale?

Spontaneous electroweak symmetry breaking (i.e. $SU(2)\times U(1)\to U(1)_{em}$ ) is at scale about 100 Gev. So, for Higgs mechanism, gauge bosons $Z$ & $W$ have masses about 100 GeV. But before ...
9
votes
2answers
407 views

Why do we seek to preserve gauge symmetries after quantization?

Gauge symmetries do not give rise to conservation laws via Noether's theorem, and they represent redundancies in our description of the system. So why do we want to keep them after quantization? For ...
9
votes
2answers
689 views

Understanding Elitzur's theorem from Polyakov's simple argument?

I was reading through the first chapter of Polyakov's book "Gauge-fields and Strings" and couldn't understand a hand-wavy argument he makes to explain why in systems with discrete gauge-symmetry only ...
9
votes
1answer
925 views

Gauge fixing and degrees of freedom

Today, my friend (@Will) posed a very intriguing question - Consider a complex scalar field theory with a $U(1)$ gauge field $(A_\mu, \phi, \phi^*)$. The idea of gauge freedom is that two solutions ...
9
votes
1answer
341 views

Invariance of Functional Integration Measure

Let us consider the functional integral: \begin{equation} \int \mathcal{D} A e^{iS[A]} \end{equation} where $S[A]$ is the action for $U(1)$ gauge field and \begin{equation} \mathcal{D}A\equiv ...
9
votes
0answers
302 views

Gauge invariant but not gauge covariant regularization

I'm not sure if someone's already asked this before, but I was wondering, in field theory, when we say that a certain field is gauge invariant but not gauge covariant, what does this mean? In ...
8
votes
3answers
822 views

Why is the Yang-Mills gauge group assumed compact and semi-simple?

What is the motivation for including the compactness and semi-simplicity assumptions on the groups that one gauges to obtain Yang-Mills theories? I'd think that these hypotheses lead to physically ...
8
votes
3answers
3k views

Noether theorem, gauge symmetry and conservation of charge

I'm trying to understand Noether's theorem, and it's application to gauge symmetry. Below what I've done so far. First, the global gauge symmetry. I'm starting with the Lagragian ...
8
votes
2answers
437 views

The gauge covariant derivative and its substitution

I was wondering wether it would make a difference (in general) if I were to were introduce the gauge covariant derivative $$D_\mu=\partial_\mu+ieA_\mu$$ In the Lagrangian density and then derive the ...
8
votes
1answer
2k views

How do I derive the Lorenz gauge from the continuity equation?

I was reading my old electromagnetics book (Elements of Electromagnetics, by Sadiku, 3rd edition) and after the author explained what the Lorenz gauge is mathematically and why it is useful in ...
8
votes
3answers
1k views

Why can't gauge bosons have mass?

Clearly, a mass term for a vector field would render the Lagrangian not gauge-invariant, but what are the consequences of this? Gauge invariance is supposed to be crucial for the renormalisation of a ...
8
votes
1answer
433 views

Introduction to Gauge Symmetries: Good, Bad or Ugly?

I'm trying to come up with a good (as in intuitive and not 'too wrong') definition of a gauge symmetry. This is what I have right now: A dynamical symmetry is a (differentiable) group of ...
8
votes
0answers
232 views

Gauge fields in Polyakov's treatment of renormalization for nonlinear sigma model

I am deriving the results of renormalization for nonlinear sigma model using Polyakov approach. I am closely following chapter 2 of Polyakov's book--- ``Gauge fields and strings''. Action for the ...
7
votes
3answers
520 views

What is the basis of gauge theory?

I’m learning about gauge concepts. I’ve always had the idea that by looking at a phenomenon from different viewpoints, that symmetries could be derived – in fact, that was what an equal sign ...
7
votes
4answers
392 views

Can an Electromagnetic Gauge Transformation be Imaginary?

The Hamiltonian of a non-relativistic charged particle in a magnetic field is $$\hat{H}~=~\frac{1}{2m} \left[\frac{\hbar}{i}\vec\nabla - \frac{q}{c}\vec A\right]^2$$. Under a gauge transformation ...
7
votes
2answers
804 views

How does non-Abelian gauge symmetry imply the quantization of the corresponding charges?

I read an unjustified treatment in a book, saying that in QED charge an not quantized by the gauge symmetry principle (which totally clear for me: Q the generator of $U(1)$ can be anything in ...
7
votes
3answers
916 views

Gauge invariant Chern-Simons Lagrangian

I have to prove the (non abelian) gauge invariance of the following lagrangian (for a certain value of $\lambda$): $$\mathcal L= -\frac14 F^{\mu\nu}_aF_{\mu\nu}^a + ...
7
votes
1answer
597 views

Why mass terms are forbidden?

I would like to clarify my understanding on why mass terms in Lagrangians of gauge theories are forbidden. It's often repeated that particle masses are forbidden by electroweak symmetry because it is ...
7
votes
1answer
159 views

Request for Reference: BRST formalism/transformations

Could anyone please suggest a very basic paper/reference/literature on BRST symmetry/formalism that requires rudimentary knowledge of Dirac's method for dealing with constrained systems and generation ...
7
votes
1answer
395 views

Some questions about Ward-Takahashi Identity

I'm a learner of Peskin and Schroeder's textbook of quantum field theory. I have proceeded to Ward-Takahashi identity and have one question when I look for Wikipedia for reference. The following is ...
7
votes
1answer
284 views

Why does local gauge invariance suggest renormalizability?

I'm reading Gauge Field Theories: An Introduction with Applications by Mike Guidry and this particular remark is not obvious to me: A tempting avenue is suggested by the QED paradigm, for if a ...
7
votes
1answer
84 views

Connection beween infinite gauge symmetries and UV finiteness

In e.g., http://arxiv.org/abs/arXiv:0712.3526 the author claims: Since the massless higher-spin field theories involve infinite-dimensional gauge symmetries, one expects that such theories may be ...
7
votes
3answers
157 views

What role does “spontaneously symmetry breaking” played in the “Higgs Mechanism”?

In talking about Higgs mechanism, the first part is always some introduction to the concept of spontaneously symmetry breaking (SSB), some people saying that Higgs mechanism is the results of SSB of ...
6
votes
2answers
269 views

Gauge invariance and Bohm-Aharonov effect

I am confused with the Bohm-Aharonov effect: though quantum mechanics is said to be gauge invariant, the presence of a solenoid imposes a gauge. I used to think that a phase shift did not change ...
6
votes
2answers
652 views

Gauge fixing and equations of motion

Consider an action that is gauge invariant. Do we obtain the same information from the following: Find the equations of motion, and then fix the gauge? Fix the gauge in the action, and then find the ...
6
votes
3answers
487 views

Why gauge theories have such a success?

[This question was inspired by a identical question asked on a other forum] Note that we may morally include general relativity in the gauge theories. We may have several (some are deliberately ...
6
votes
2answers
745 views

Intuition for gauge parallel transport (Wilson loops)

I'm looking for a geometrical interpretation of the statement that "Wilson loop is a gauge parallel transport". I have seen QFT notes describe U(x,y) as "transporting the gauge transformation", and ...
6
votes
1answer
109 views

“gauge fixed world-sheet action”

My question is in reference to the action in equation 4.130 of Becker, Becker and Schwartz. It reads as, $S_{matter}= \frac{1}{2\pi}\int (2\partial X^\mu \bar{\partial}X_\mu + \frac{1}{2}\psi^\mu ...
6
votes
1answer
291 views

Noether's identities

I have some questions about the Noether's second theorem (generally not covered by field theory books): What is the most general Noether identity for (classical) field theories? Why are Noether ...
6
votes
1answer
232 views

What do people mean by gauge invariance of the normalization of field?

Lets have the scalar Klein-Gordon field interacting with EM field: $$ L = \partial_{\mu}\varphi \partial^{\mu}\varphi - m^2\varphi \varphi^{*} - j_{\mu}A^{\mu} + q^{2} A_{\mu}A^{\mu}\varphi ...
6
votes
1answer
298 views

How do non-transverse photon polarizations cancel in Euclidean QED?

First, recall how to write scattering amplitudes in covariant fashion in Minkowskian QED. One starts by considering some process with an external photon whose momentum is chosen to be ...
6
votes
0answers
137 views

Faddeev Popov Gauge Fixing in Electromagnetism

Reading section 9.4 in Peskin, I am wondering about the following: The functional integral on $A_{\mu}$ diverges for pure-gauge configurations, because for those configurations, the action is zero. ...
6
votes
0answers
317 views

gauge invariance of the Feynman amplitudes

When we calculate the photon polarization sums over amplitudes, $$X=\sum\limits_{r=1}^{2}|\mathcal M_r|^2=\mathcal M_\alpha\mathcal M_\beta^*\sum\limits_{r=1}^{2}\epsilon_r^\alpha\epsilon_r^\beta$$ ...
5
votes
2answers
197 views

When can we add a total time derivative of $f(q, \dot{q}, t)$ to a Lagrangian?

The other day, I was listening to this lecture on the Lagrangian for a charged particle in an electromagnetic field, and at one point in the video, the lecturer mentions that we can add any total time ...
5
votes
1answer
394 views

Field strength vanishes iff $A_{\mu}$ is pure gauge

Is it true that the field strength $F_{\mu\nu}$ in a non-Abelian gauge theory with gauge group $G$ vanishes if, and only if, the gauge field $A_{\mu}$ is a pure gauge? I can show one implication. ...
5
votes
2answers
232 views

Electric current $j^{\mu}$ in standard QED vs. scalar QED

The expression for the 4-current $j^{\mu}$ in standard QED is $$ e\bar{\Psi}\gamma^\mu\Psi $$ and $$ \frac{e}{2 i}(\psi^\dagger D^\mu \psi - (D^\mu \psi)^\dagger \psi) $$ in scalar QED. I ...
5
votes
1answer
434 views

6 independent Einstein field equations?

I can't understand the comment on page 409, Gravitation, by Misner, Thorne, Wheeler It follows that the ten components $G_{\alpha\beta} =8\pi T_{\alpha\beta}$ of the field equation must not ...
4
votes
2answers
663 views

Why is the “canonical momentum” for the Dirac equation not defined in terms of the “gauge covariant derivative”?

The canonical momentum is always used to add an EM field to the Schrödinger/Pauli/Dirac equations. Why does one not use the gauge covariant derivative? As far as I can see, the difference is a factor ...
4
votes
3answers
490 views

Problems with putting mass on Yang-Mills theory by hand

When Yang-Mills field theory was introduced, a problem is that the gauge invariance can not allow mass for the gauge field. Later people invented spontaneous symmetry breaking and Higgs mechanism to ...
4
votes
1answer
157 views

$SU(2)$ gauge symmetry

Take the Lagrangian with one fermion: $$ \mathcal{L} = -\frac{1}{4}F^{\mu\nu}_aF^a_{\mu\nu} + \bar{\psi}(i\gamma^\mu D_\mu - m)\psi$$ where the gauge covariant derivative $D_\mu = ...
4
votes
1answer
107 views

$U(1){\times}U(1)$ local gauge invariance derivative

In QED and the basic Higgs mechanism, there is a local gauge transformation where a scalar field $\phi$ is transformed as: $e^{i\theta\eta(x)} \phi$ The partial derivative of this however makes the ...
4
votes
1answer
347 views

What exactly is the connection between gauge transformations and symmetry groups?

For a given gauge transformation, say, the electromagnetic field, where observable quantities aren't affected by transformations of the form $$\mathbf{A}' = \mathbf{A} + \nabla \chi,$$ $$\phi' = \phi ...